On Coloring the Arcs of Biregular Graphs

Abstract

Recalling each edge of a graph $H$ has 2 oppositely oriented arcs, each vertex $v$ of $H$ is identified with the set of arcs, denoted $(v,e)$, departing from $v$ along the edges $e$ of $H$ incident to $v$. Let $H$ be a $(\lambda,\mu)$-biregular graph with bipartition $(Y,X)$, where $|Y|=k\mu$ and $|X|=k\lambda$, ($0<k,\lambda,\mu\in\mathbb{Z}$). We consider the problem, for each edge $e=yx$ in $H$, of assigning, a color (given by an element) of $Y$, resp. $X$, to the arc $(y,e)$, resp. $(x,e)$, so that each color is assigned exactly once in the set of arcs departing from each vertex of $H$. Furthermore, we set such assignment to fulfill a specific bicolor weight function over a monotonic subset of $Y\times X$. This problem applies to the Design of Experiments for Industrial Chemistry, Molecular Biology, Cellular Neuroscience, etc. An algorithmic construction based on biregulzr graphs with bipartitions given by cyclic-group pairs is presented, as well as 3 essentially different solutions to the Great Circle Challenge Puzzle based on a different biregular graph whose bipartition is formed by the vertices and 5-cycles of the Petersen graph.